Non-standard behavior of density estimators for sums of squared observations

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Date
2008
Volume
2008-07
Issue
Journal
Series Titel
Oberwolfach Preprints (OWP)
Book Title
Publisher
Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach
Link to publishers version
Abstract

It has been shown recently that, under an appropriate integrability condition, densities of functions of independent and identically distributed random variables can be estimated at the parametric rate by a local U-statistic, and a functional central limit theorem holds. For the sum of two squared random variables, the integrability condition typically fails. We show that then the estimator behaves differently for different arguments. At points in the support of the squared random variable, the rate of the estimator slows down by a logarithmic factor and is independent of the bandwidth, but the asymptotic variance depends on the rate of the bandwidth, and otherwise only on the density of the squared random variable at this point and at zero. A functional central limit theorem cannot hold. Of course, for bounded random variables, the sum of squares is more spread out than a single square. At points outside the support of the squared random variable, the estimator behaves classically. Now the rate is again parametric, the asymptotic variance has a different form and does not depend on the bandwidth, and a functional central limit theorem holds.

Description
Keywords
Convolution density estimator, smoothness of convolutions, asymptotically linear estimator
Citation
Schick, A., & Wefelmeyer, W. (2008). Non-standard behavior of density estimators for sums of squared observations (Vol. 2008-07). Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach. https://doi.org//10.14760/OWP-2008-07
License
This document may be downloaded, read, stored and printed for your own use within the limits of § 53 UrhG but it may not be distributed via the internet or passed on to external parties.
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